It is not H’s organization. He was credited with the idea for the Games. He’s also currently a patron of the Games but has nothing to do with the actual management or running of the Games or the Invictus Foundation. In theory they would use his name and appearance at events to get publicity and shine more attention on the games and the veterans who participate in the events but in practice TW has turned the last two Games into her own ridiculous fashion show. She’s tried to make it all about her instead of the veterans. The Royal Foundation which originally involved William, Catherine, and Harry, donated a substantial sum to jumpstart the first Invictus Games in 2014. William also donated the 1 million BP settlement he received from the newspapers over the phone hacking scandal to the Invictus Foundation. TW briefly joined the Royal Foundation after she married H but that obviously proved to be an untenable situation and the Harkles left the Royal Foundation to do their own thing in 2019. I don’t believe H or TW or any of the entities they set up after the split with the Royal Foundation ever donated any funds to the Invictus Foundation or the Games.
Edit: the Invictus Games was actually patterned after the U.S. Warrior Games which started in 2010. Some people have suggested that H’s handlers gave him the idea to sponsor the Invictus Games to give him something to do after he left the military.